Costelloe moves to Wigan in a permanent transfer and for an undisclosed fee.
The 22-year-old joined the Clarets from Galway United in 2021 and after progressing through the U21s, the Irishman made his senior debut for the Club in the Championship in July 2022.
Costelloe has spent time out on loan at Bradford City, St Johnstone, Dundee, Accrington Stanley and Northampton Town.
He told Wigan's website: “I’m delighted to get the move over the line. It’s been a great day for me and my family - I brought my Dad over with me, and having seen the stadium, we definitely made the right choice.
“It is a great Club, and I cannot wait to get started. I’ve spoken to the Gaffer for quite a bit, and the plan that was outlined is to get this Club back to the Championship. That’s my ambition as well, so we’ll try to achieve that.
“I’ve seen the strikers the Gaffer has worked with, and how he’s helped progress their careers. Hopefully, he can do the same with me."